To find the solution(s) to the equation [tex]\(x^2 - 4x + 4 = 2x + 1 + x^2\)[/tex], we can use the graph of two functions. The steps are as follows:
1. Rewrite the equation in a standard form:
We start with the equation:
[tex]\[
x^2 - 4x + 4 = 2x + 1 + x^2
\][/tex]
2. Simplify the equation:
Combine like terms on both sides of the equation. Subtract [tex]\(x^2\)[/tex] from both sides:
[tex]\[
x^2 - 4x + 4 - x^2 = 2x + 1 + x^2 - x^2
\][/tex]
Simplifying this gives:
[tex]\[
-4x + 4 = 2x + 1
\][/tex]
3. Further simplify to isolate the variable [tex]\(x\)[/tex]:
Move all the terms involving [tex]\(x\)[/tex] to one side and constant terms to the other side:
[tex]\[
-4x - 2x = 1 - 4
\][/tex]
Combine like terms:
[tex]\[
-6x = -3
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides by [tex]\(-6\)[/tex]:
[tex]\[
x = \frac{-3}{-6}
\][/tex]
Simplify the fraction:
[tex]\[
x = \frac{1}{2}
\][/tex]
5. Interpret the solution graphically:
To confirm this result graphically, you can plot the functions:
[tex]\[
y_1 = x^2 - 4x + 4
\][/tex]
and
[tex]\[
y_2 = 2x + 1 + x^2
\][/tex]
Graph these two functions on the same coordinate plane. The solution to the equation [tex]\(x^2 - 4x + 4 = 2x + 1 + x^2\)[/tex] is the [tex]\(x\)[/tex]-value where the graphs of [tex]\(y_1\)[/tex] and [tex]\(y_2\)[/tex] intersect.
Since the solution as calculated is [tex]\(x = \frac{1}{2}\)[/tex], the graphs of the two functions will intersect at [tex]\(x = \frac{1}{2}\)[/tex]. Therefore, the point of intersection corresponds to [tex]\(x = \frac{1}{2}\)[/tex] and confirms our solution.
